---
title: "Maurer-Cartan"
aliases: [Maurer-Cartan]
created: 2022-04-05T23:42
updated: 2023-04-14T16:56
---


---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [DGLAs](Unsorted/DGA.md)
	- [Artinian algebra](Artinian%20algebra.md)
	- [deformation theory](deformation%20theory.md)

---

# Maurer-Cartan

Motivation:

![](2023-03-31-13.png)

![](2023-03-31-14.png)

![](attachments/Pasted%20image%2020220213204328.png)

![](attachments/Pasted%20image%2020220525152910.png)

![](attachments/Pasted%20image%2020220525152806.png)

![](attachments/Pasted%20image%2020220525152834.png)

# Formal solutions

Look for formal power series solutions $\xi(t) = \sum \xi_i t^i$ with $\xi_i \in A^{0, 1}(M, \T_{1, 0})$ to the MC equation to get a [Kodaira-Spencer](Kodaira-Spencer.md) class:

![](2023-03-31-22.png)

Equivalent deformations have the same KS class.

![](2023-03-31-23.png)

Particularly nice in the [CY](Calabi-Yau%20manifold.md) case:
![](2023-03-31-24.png)

# The deformation functor

![](2023-03-31-18.png)

# Complex structures

Relation to [integrable almost complex structures](integrable%20almost%20complex%20structures):

![](2023-03-31-21.png)

# Explicit motivation

![](2023-04-02-23.png)